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Non-Euclidean Geometry

  The Elements of Euclid, formulated in the 3rd century  BCE, for almost 2,000 years seemed to be the last word in geometry; they gave a system of axiomatization for geometry together with a large number of theorems proved using just these axioms. But one axiom, the parallel axiom (stating that given a line L and a point a not on L there is a unique line through a which is parallel to L i.e. not intersecting it everywhere) did not seem in the same realm of obviousness as the others, especially as a pair of parallel lines (for example, railway tracks) do appear to meet on the horizon. For this reason, mathematicians spent many years trying to prove that the parallel axiom was a consequence of the other axioms. These attempts led directly to the development of non-Euclidean geometry, because researchers began to try and prove it using proof by contradiction, which meant that they were assuming that the axiom was not true and were trying to derive a contradiction from this.

Although none was ever found, it was not until the early 19th century that anyone was daring enough to suggest that this meant that a new kind of geometry had been discovered in which the parallel axiom was not true. Johann Bolyai (1802 - 1860) and , Nicholas Lobachevsky (1792 - 1856) were the two obscure mathematicians who first had the courage to put this idea forward, to establishment derision. It gradually became clear that they were right, and non-Euclidean geometry was established on an equal footing with the Euclidean variety.

Non-Euclidean geometry may appear at first only to be a mental exercise, something that is purely a construct of the imagination. After all, space seems Euclidean and engineering from start to finish. In architecture, Euclidean geometry is used with great success in the everyday, and that would seem to be convincing proof.

One of the major predictions of non-Euclidean geometry is that no triangle has the sum of its angles being 180°, whereas Euclidean geometry says that all triangles have the sum of their angles being exactly that figure. , Karl Friedrich Gauss (1777 - 1832) set out to show that the universe was Euclidean, by shining lights between three mountain peaks several miles apart and measuring the angles between them. He discovered that the triangle thus formed was Euclidean up to the limits of his measuring devices. The problems with this approach are that the defect of the triangle could be too small to measure (so that it is never possible positively to prove that the universe is Euclidean by this method), that the tools used assume in their design a Euclidean geometry (so Gauss could have seen exactly what he expected to see), and that it is dependent on the assumption that light rays travel in straight lines. The modern view, the general relativity of Einstein (1879 - 1955), is that light does indeed travel in curved paths which are dependent on the distribution of matter in space. Central to general relativity is the assumption that the universe is non-Euclidean in its geometry (though it approximates to a Euclidean geometry on a local scale). The type of non-Euclidean geometry used by general relativity is more complex than the non-Euclidean geometry of Bolyai and Lobachevsky. Geometry, though, is a mathematical discipline, and so does not need to mirror the real world; to insist that it did so would be to turn it into an experimental science. Instead, we can choose whatever version of geometry we want, whichever is best suited for the purpose for which we wish to use it—Euclidean for engineering and architecture, for example, and non-Euclidean for general relativity.

The destruction of Euclidean geometry as absolute truth was instrumental in bringing about major changes in the philosophy of mathematics. Until the mid-19th century, mathematical truth was assumed to be absolute, a priori truth; Euclidean geometry was, for example, held up by Kant as the paradigm of this view of mathematics. Once this idea was shattered, the way was paved for new schools of thought about what mathematics really is: formalism, intuitionism and logicism. Non-Euclidean geometry had a particular influence on formalism; once it was realized that the words ‘points’ and ‘lines’ could be replaced in the axioms of geometry by any other pair of words which were not tied down by intuitive ideas of points and lines, and that the theorems were still valid deductions from these axioms (because their derivation did not depend on the diagrams used to make them intuitively plausible in textbook demonstrations), it was only a short step to deciding that all of mathematics consisted merely of the formal manipulations of symbols. SMcL



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